Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11284 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #397 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1442 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,15,12,16,10,25)(8,17,19,13,20,29,26,28)(9,21,24,36,23,14,22,30)(18,32,33,35,34,31,37,27);(6,8)(7,13)(9,18,23,34)(10,26)(11,19)(12,20)(14,27,21,35)(15,29)(16,28)(17,25)(22,37,24,33)(30,32,36,31);(6,9,10,22,12,23,11,24)(7,14,15,30,16,21,25,36)(8,18,19,33,20,34,26,37)(13,27,17,31,28,35,29,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4272);